Slightly expanded version of this post now available on The Guardian.

The conference on “Supersymmetry and the Unification of Fundamental Interactions“, which my colleagues and I organised in Bonn, finished yesterday. The entire week I was thinking I would drop into bed and sleep for a full day. But oddly, I feel quite refreshed. It was great fun listening to the talks and discussing with so many friends and colleagues, despite all the organisational headaches. The conference dinner was on an elegant boat which in an earlier life was used for the signing ceremony of the Schengen agreement. (For us mainland Europeans this is a big deal.)

Supersymmetry seems alive and well and ready to face the challenge from the LHC. But what is supersymmetry? And what is so super about it? Why are we so taken with it, even though there is as yet no experimental evidence it actually exists? There are two main arguments, first it is a solution to the `hierarchy problem’. I will save this for a potential second post, if Jon invites me back. The other is indeed an aesthetic argument related to the `Coleman-Mandula theorem’.

Now, I tell myself every morning in front of the bathroom mirror, that aesthetics is for wimps, but it is all the same an interesting argument.

Symmetries have become a central pillar of our understanding of nature. A sphere is symmetric in the sense that if you leave me in a room with the sphere and come back in, you can not tell if and by possibly how much and about which axis I have rotated the sphere. The sphere is highly symmetric. This however also makes a sphere kind of boring, since it has to be the same in every direction it has no structure. If the sphere has a pattern on it, like for example an old black and white football, only very specific rotations are still undetectable, this is the remaining, reduced symmetry.

It turns out that in the world of elementary particles there are two types of symmetries. One kind are internal symmetries. These govern the forces of nature like the electromagnetic force. Here a hidden, internal, property of particles is changed. The other kind we call external symmetries and they affect the way particles fly through space and time. The appropriate external symmetry is described by special relativity, invented by Einstein in 1905. The undetectable transformations are called Lorentz transformations. In this case the laws of nature are unchanged if we look at them for example on a stationary train or one moving with constant velocity (and on smooth tracks!).

Now how about Coleman and Mandula? They showed that in fact the Lorentz symmetries are the maximal external symmetry allowed in nature. If you were to introduce a larger more extensive symmetry the world would become so boring that particles could no longer interact. They would just fly around freely in space not knowing about each other. In their argument Coleman and Mandula however neglected one external property of particles, their spin. This is a peculiar quantum property, they behave as if they had a small internal magnet. In specific units all the matter particles we know, e.g. the electron and the quarks have spin 1/2. The force carriers like the photon have spin 1. Spin is an external property, which is affected by rotations in space. Now if we extend Coleman and Mandula and allow for discrete changes of spin by half a unit, we find a new maximal external symmetry of nature. This is supersymmetry. It is super because it goes beyond the previous external symmetries. If nature is supersymmetric the electron must have a partner with spin 0 and the photon a partner with spin 1/2 and all with many interactions.

However, if this symmetry were at all extended (taking now also spin into account, of course) the resulting world would be boring and trivial with no interactions. Since we have now used up all external particle properties we believe this is the end of the line. This is what makes supersymmetry so special….and to some beautiful.